Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Geometric and Topological Data Analysis: Big Picture, Lecture notes of Probability and Statistics

The estimation of geometric and topological features of probability density functions. The classical approach of using plug-in estimates from the kernel density estimator (KDE) is presented, but it may fail. The failure of KDE in analyzing data is also discussed. mathematical formulas and graphs.

Typology: Lecture notes

2022/2023

Uploaded on 05/11/2023

amlay
amlay 🇺🇸

4.1

(19)

384 documents

1 / 74

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
GEOMETRIC AND TOPOLOGICAL DATA
ANALYSIS
Yen-Chi Chen
Department of Statistics
University of Washington
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
pf12
pf13
pf14
pf15
pf16
pf17
pf18
pf19
pf1a
pf1b
pf1c
pf1d
pf1e
pf1f
pf20
pf21
pf22
pf23
pf24
pf25
pf26
pf27
pf28
pf29
pf2a
pf2b
pf2c
pf2d
pf2e
pf2f
pf30
pf31
pf32
pf33
pf34
pf35
pf36
pf37
pf38
pf39
pf3a
pf3b
pf3c
pf3d
pf3e
pf3f
pf40
pf41
pf42
pf43
pf44
pf45
pf46
pf47
pf48
pf49
pf4a

Partial preview of the text

Download Geometric and Topological Data Analysis: Big Picture and more Lecture notes Probability and Statistics in PDF only on Docsity!

Geometric and Topological Data

Analysis

Yen-Chi Chen

Department of Statistics

University of Washington

Geometric and Topological Data Analysis: Big Picture

Geometric and Topological Data Analysis: Big Picture

The data can be viewed as

X

, · · · , X

n

∼ p,

p is a probability density function.

Scientists are interested in geometric

or topological features of p.

Geometric and Topological Data Analysis: Big Picture

The data can be viewed as

X

, · · · , X

n

∼ p,

p is a probability density function.

Scientists are interested in geometric

or topological features of p.

Geometric and Topological Data Analysis: Big Picture

The data can be viewed as

X

, · · · , X

n

∼ p,

p is a probability density function.

Scientists are interested in geometric

or topological features of p.

Geometric and Topological Data Analysis: Big Picture

The data can be viewed as

X

, · · · , X

n

∼ p,

p is a probability density function.

Scientists are interested in geometric

or topological features of p.

The Classical Approach

◦ In all the above examples, how we estimate the

geometric/topological structures is based on plug-in estimates

from the kernel density estimator (KDE).

◦ Namely, we estimate the probability density function first and

then convert it into an estimator of the corresponding structure.

The Classical Approach

◦ In all the above examples, how we estimate the

geometric/topological structures is based on plug-in estimates

from the kernel density estimator (KDE).

◦ Namely, we estimate the probability density function first and

then convert it into an estimator of the corresponding structure.

◦ But this idea may fail.

Failure of KDE in Analyzing Data

Density Ranking: Introduction

◦ The KDE cannot detect intricate structures inside the GPS data.

◦ But the density ranking works!

Density Ranking: Introduction

◦ The KDE cannot detect intricate structures inside the GPS data.

◦ But the density ranking works!

◦ This comes from the fact that the underlying probability density

function (PDF) does not exist!

◦ Namely, our probability distribution function is a singular

measure.

Density

  • 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.
  • −0.
  • −0.
  • −0.
  • −0.
  • −0.
  • −0.
  • −0.
  • −0.
    • 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0. Failure of KDE in Analyzing Data
  • −0.
  • −0.
  • −0.
  • −0.
  • −0.
  • −0.
  • −0.
  • −0.
  • Definition of Density Ranking - - −0.2 0.0 0.2 0.4 0.6 0.8 1.

Definition of Density Ranking - 3

◦ The density ranking is a transformed quantity from the KDE.

◦ Instead of using the density value, we focus on the ranking of it.